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\begin{equation*}\frac{d}{dx}(e^x e^{\mathbf{i} y}) = (\frac{d}{dx} e^x) e^{\mathbf{i} y} + e^x (\frac{d}{dx} e^{\mathbf{i} y}) = e^{x + \mathbf{i} y} + e^x (\frac{d}{dx} e^{\mathbf{i} y}\end{equation*}
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